Evaluate the Integral

Problem

(∫_1^2)((e1/(x3))/(x4)*d(x))

Solution

1. Identify a suitable substitution to simplify the exponent. Let u=1/(x3) which can be written as u=x(−3)

2. Differentiate u with respect to x to find the differential d(u)

d(u)/d(x)=−3*x(−4)

d(u)=−3/(x4)*d(x)

−1/3*d(u)=1/(x4)*d(x)

3. Change the limits of integration to correspond with the variable u
   When x=1 u=1/1=1
   When x=2 u=1/2=1/8

4. Substitute the new variable and limits into the integral.

(∫_1^1/8)(eu*(−1/3)*d(u))

5. Simplify the integral by pulling out the constant and reversing the limits to remove the negative sign.

1/3*(∫_1/8^1)(eu*d(u))

6. Evaluate the definite integral using the fundamental theorem of calculus.

1/3*[eu]1(1/8)

1/3*(e1−e(1/8))

Final Answer

(∫_1^2)((e1/(x3))/(x4)*d(x))=(e−√(8,e))/3

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